Our proof relies on the construction of the SIR-\(\omega \)  model given in [1]}. We will give the details in the next section. Using that construction we can derive evolution equations for quantities like \(S^{(n)}(t)\) .
As pointed out in [1]}, a common approach to analyze epidemic models is performing a suitable time-transform so that the system becomes more solvable. In the case of the SIR-\(\omega \)  model, after the time-change, the limiting system of ODEs change from (REF )-() to (REF )-(). Note that the equation for \(\widehat{i}^{\prime }\)  involves the term \(\widehat{i}/\widehat{i}_E\) , which is not Lipschitz for \(\widehat{i}_E\)  near 0.
Thus one cannot directly apply classical results (e.g., [3]}) to obtain the convergence of the terminal time \(\widehat{\tau }^{(n)}\)  and the fraction of infected vertices \(T^{(n)}/n\) .
